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feat: addition elab and node identities

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jstoobysmith 2024-10-22 11:49:58 +00:00
parent ecb2c7778c
commit 6fbace33da
5 changed files with 165 additions and 70 deletions
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11
HepLean/Tensors/ComplexLorentz/Lemmas.lean
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@ -115,7 +115,7 @@ lemma coMetric_0_0_field : {Lorentz.coMetric | 0 0}ᵀ.field = 1 := by
set_option maxRecDepth 20000 in
lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
{T2 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V} :
{(T1 | μ ν ⊗ T2 | μ ν) = (T2 | μ ν ⊗ T1 | μ ν)}ᵀ := by
{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
rw [perm_tensor_eq (contr_tensor_eq (contr_tensor_eq (prod_comm _ _ _ _)))]
rw [perm_tensor_eq (contr_tensor_eq (perm_contr _ _))]
rw [perm_tensor_eq (perm_contr _ _)]
@ -134,7 +134,7 @@ lemma contr_rank_2_symm {T1 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
lemma contr_rank_2_symm' {T1 : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
{T2 : (Lorentz.complexContr ⊗ Lorentz.complexContr).V} :
{(T1 | μ ν ⊗ T2 | μ ν) = (T2 | μ ν ⊗ T1 | μ ν)}ᵀ := by
{T1 | μ ν ⊗ T2 | μ ν = T2 | μ ν ⊗ T1 | μ ν}ᵀ := by
rw [perm_tensor_eq contr_rank_2_symm]
rw [perm_perm]
rw [perm_eq_id]
@ -181,6 +181,13 @@ lemma symm_contr_antiSymm {S : (Lorentz.complexCo ⊗ Lorentz.complexCo).V}
rw [contr_rank_2_symm', perm_tensor, antiSymm_contr_symm hA hs]
rfl
lemma antiSymm_add_self {A : (Lorentz.complexContr ⊗ Lorentz.complexContr).V}
(hA : {A | μ ν = - (A | ν μ)}ᵀ) :
{A | μ ν + A | ν μ}ᵀ.tensor = 0 := by
rw [← TensorTree.add_neg (twoNodeE complexLorentzTensor Color.up Color.up A)]
apply TensorTree.add_tensor_eq_snd
rw [neg_tensor_eq hA, neg_tensor_eq (neg_perm _ _), neg_neg]
end Fermion
end